Pipe Systems: Series, Parallel and Networks
The system curve, pipes in series and parallel, branching networks, Hardy Cross, and choosing a pipe size and velocity.
The system curve
A pump does not push a fixed pressure: it supplies whatever head the system demands at that flow. The relation between flow and the head the pipework needs is the system curve. It has a fixed part, the static head (lift plus any pressure difference between the ends), and a part that grows with the square of the flow: friction and fittings.
Hsystem(Q) = Hstatic + (f L ÷ D + ΣK) v² ÷ 2g
Worked example: system curve of the DN80 line
Lift 12 m, 100 m of DN80 steel pipe, fittings ΣK = 8.45 (from Fittings, Valves and Minor Losses). Head needed at each flow:
| Flow | Velocity | Static | Pipe friction | Fittings | System head |
|---|---|---|---|---|---|
| 0 L/s | 0.00 m/s | 12.0 m | 0.00 m | 0.00 m | 12.0 m |
| 5 L/s | 1.05 m/s | 12.0 m | 1.52 m | 0.47 m | 14.0 m |
| 10 L/s | 2.10 m/s | 12.0 m | 5.63 m | 1.89 m | 19.5 m |
| 15 L/s | 3.15 m/s | 12.0 m | 12.24 m | 4.26 m | 28.5 m |
| 20 L/s | 4.19 m/s | 12.0 m | 21.34 m | 7.58 m | 40.9 m |
| 25 L/s | 5.24 m/s | 12.0 m | 32.93 m | 11.84 m | 56.8 m |
| 30 L/s | 6.29 m/s | 12.0 m | 47.00 m | 17.05 m | 76.0 m |
At zero flow the head is just the 12 m of lift. As flow rises the friction and fittings take over: at 30 L/s they are 64.0 m, more than the lift. This is why throttling a pump (adding a valve K) makes the curve steeper and moves the operating point.
Pipes in series
In series the flow is the same in each pipe and the head losses add: h = h₁ + h₂ + …
Worked example: two sizes in series
- 10 L/s flows through 60 m of DN80 (ID 77.92 mm) then 40 m of DN50 (ID 52.50 mm).
- Losses: h₁ = 3.38 m, h₂ = 16.76 m, total 20.14 m (197.2 kPa). The 40 m of narrow pipe lose 83 percent of the total although it is only 40 percent of the length: the smallest pipe dominates.
Pipes in parallel
In parallel the head loss is the same across each branch (they connect the same two points) and the flows add: Q = Q₁ + Q₂ + … Since h grows with Q², the flow splits in proportion to √(1 ÷ r), where r is the resistance of each branch (h = r Q²): the bigger pipe carries far more.
Worked example: two parallel pipes
- 20 L/s has to be shared by a DN80 and a DN50 pipe, each 100 m long, between the same two tanks. Find the split such that both have the same head loss.
- Solving (by bisection on Q₁): Q₁ = 14.79 L/s through the DN80 (3.10 m/s) and Q₂ = 5.21 L/s through the DN50 (2.41 m/s), with a common head loss of 11.92 m.
- The DN80 pipe, with only 48 percent more diameter, carries 2.8 times the flow of the DN50: the carrying capacity of a pipe goes as about D2.6 at constant gradient.
Networks and the Hardy Cross method
For a loop or network, the unknown flows are found by two rules: (1) flow in = flow out at each junction and (2) the head losses around any closed loop add to zero. The Hardy Cross method guesses flows that satisfy rule 1, then corrects them around each loop until rule 2 is met:
ΔQ = − Σ r Q|Q| ÷ (2 Σ r |Q|)
Worked example: Hardy Cross on the parallel pair
Start from the guess Q₁ = 15 L/s, Q₂ = 5 L/s (total 20 L/s). Resistances r₁ = 54 451 and r₂ = 439 515 s²/m⁵ from the friction factors above.
| Pass | Loop imbalance | Correction ΔQ | Q₁ | Q₂ |
|---|---|---|---|---|
| 1 | 1.2637 m | -0.210 L/s | 14.790 L/s | 5.210 L/s |
| 2 | -0.0169 m | 0.003 L/s | 14.793 L/s | 5.207 L/s |
| 3 | 0.0000 m | 0.000 L/s | 14.793 L/s | 5.207 L/s |
Three passes reach the direct solution (14.79 and 5.21 L/s). Real networks do the same on several loops at once, and software such as EPANET does it automatically.
Choosing a pipe size and velocity
A small pipe is cheap to buy but costs pressure, and so pumping energy, every hour it runs; a large pipe costs more to buy and install but wastes less energy. The best size minimizes the total over the life of the system, which in practice means keeping the velocity in a sensible range.
| Service | Typical velocity |
|---|---|
| Pump suction (short, liquid near boiling) | 0.5 to 1.5 m/s |
| Pump discharge, water | 1.5 to 3 m/s |
| Water mains and building services | 1 to 2 m/s |
| Hydraulic oil, pressure lines | 3 to 6 m/s |
| Compressed air, headers | 6 to 10 m/s |
| Steam, low pressure | 20 to 40 m/s |
Worked example: which size for 10 L/s over 100 m?
| Pipe | Inside diameter | Velocity | Pressure drop | Pump input power for friction | Energy per year (8 000 h) |
|---|---|---|---|---|---|
| DN50 | 52.5 mm | 4.62 m/s | 410.2 kPa | 5 860 W | 46 884 kWh |
| DN65 | 62.7 mm | 3.24 m/s | 165.9 kPa | 2 371 W | 18 964 kWh |
| DN80 | 77.9 mm | 2.10 m/s | 55.1 kPa | 787 W | 6 299 kWh |
| DN100 | 102.3 mm | 1.22 m/s | 14.1 kPa | 202 W | 1 614 kWh |
| DN125 | 128.2 mm | 0.77 m/s | 4.6 kPa | 66 W | 527 kWh |
DN80 (about 2 m/s) is a reasonable choice: dropping to DN65 doubles the friction energy, while going up to DN100 saves little. The saving from a larger pipe has to be set against its purchase price, fittings, supports and the price of electricity: the Pump Power and Sizing Calculator includes the energy cost.
Related
- Previous: Fittings, Valves and Minor Losses. Next: Pumps, Pump Curves and NPSH.
- Pipe Flow and Pressure Drop Calculator.
- Pump Power and Sizing Calculator.
- Fluid Mechanics and Pipe Flow: the whole guide.
FAQ
Why does the flow prefer the bigger pipe so strongly?
Because friction loss at a given flow goes roughly as 1 ÷ D⁵, a pipe that is 50 percent wider has about 8 times less resistance, so it takes most of the flow at the same head loss.
Does adding a parallel pipe always help?
It lowers the system resistance, so the flow rises, but a pump will move along its curve to a higher flow and a lower head. Check that the pump is not pushed into the high-flow, cavitating end of its curve.
What about branches that open and close?
Then the system curve changes with the valve positions. Calculate the worst case (all open, and the longest branch) for the pump, and use balancing valves where several branches must share flow in set proportions.
Can I use Hardy Cross by hand?
For one or two loops, yes, in a spreadsheet. For larger networks use software such as EPANET (free), which solves them directly.
References
- Cross, H., “Analysis of flow in networks of conduits or conductors”, University of Illinois Bulletin 286, 1936.
- Hydraulic Institute, Pipe Friction Manual and ANSI/HI 9.6.7 (pump system effects).
- Rossman, L. A., EPANET 2 Users Manual, US EPA.
- White, F. M., Fluid Mechanics, McGraw-Hill (pipe networks).
Last reviewed: October 2026. Costs and velocities are typical.